A169942 -id:A169942 - cp's OEIS frontend

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.

This is a kind of completeness property, cf. A126796.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
             (111)  (1111)  (2111)   (3111)    (4111)     (32111)
                            (11111)  (21111)   (22111)    (41111)
                                     (111111)  (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If weak run-sums are distinct we have A353865, the completion of A353864.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, comps A353860.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768.
Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]

\\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b,b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024

a(31) onwards from Andrew Howroyd, Jan 15 2024

A143824 Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset: 0

John W. Layman, Sep 02 2008

When the set {x(1),x(2),...,x(k)} satisfies the property that all differences |x(i)-x(j)| are distinct (or alternately, all the sums are distinct), then it is called a Sidon set. So a(n) is the maximum cardinality of a dense Sidon subset of {1,2,...,n}. - Sayan Dutta, Aug 29 2024
See A143823 for the number of subsets of {1, 2, ..., n} with the required property.
See A003022 (and A227590) for the values of n such that a(n+1) > a(n). - Boris Bukh, Jul 28 2013
Can be formulated as an integer linear program: maximize sum {i = 1 to n} z[i] subject to z[i] + z[j] - 1 <= y[i,j] for all i < j, sum {i = 1 to n - d} y[i,i+d] <= 1 for d = 1 to n - 1, z[i] in {0,1} for all i, y[i,j] in {0,1} for all i < j. - Rob Pratt, Feb 09 2010
If the zeroth term is removed, the run-lengths are A270813 with 1 prepended. - Gus Wiseman, Jun 07 2019

			For n = 4, {1, 2, 4} is a subset of {1, 2, 3, 4} with distinct differences 2 - 1 = 1, 4 - 1 = 3, 4 - 2 = 2 between pairs of elements and no larger set has the required property; so a(4) = 3.
From _Gus Wiseman_, Jun 07 2019: (Start)
Examples of subsets realizing each largest size are:
   0: {}
   1: {1}
   2: {1,2}
   3: {2,3}
   4: {1,3,4}
   5: {2,4,5}
   6: {3,5,6}
   7: {1,3,6,7}
   8: {2,4,7,8}
   9: {3,5,8,9}
  10: {4,6,9,10}
  11: {5,7,10,11}
  12: {1,4,5,10,12}
  13: {2,5,6,11,13}
  14: {3,6,7,12,14}
  15: {4,7,8,13,15}
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..500
Thomas Bloom, Problem 30, Problem 43, Problem 155, and Problem 861, Erdős Problems.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
Terence Tao, Erdős problem database, see nos. 30, 43, 155, 861.
Wikipedia, Sidon sequence.
Wikipedia, Golomb ruler
Balogh, J., Füredi, Z., & Roy, S. (2023), An Upper Bound on the Size of Sidon Sets, The American Mathematical Monthly, 130(5), 437-445.

Cf. A143823, A003002, A003022, A227590.

Cf. A108917, A169942, A270813, A325676, A325677, A325678, A325683, A325687.

Table[Length[Last[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[#,{2}]&]]],{n,0,15}] (* Gus Wiseman, Jun 07 2019 *)

For n > 1, a(n) = A325678(n - 1) + 1. - Gus Wiseman, Jun 07 2019
From Sayan Dutta, Aug 29 2024: (Start)
a(n) < n^(1/2) + 0.998*n^(1/4) for sufficiently large n (see Balogh et. al. link).
It is conjectured by Erdos (for $500) that a(n) < n^(1/2) + o(n^e) for all e>0. (End)

More terms from Rob Pratt, Feb 09 2010
a(41)-a(60) from Alois P. Heinz, Sep 14 2011
More terms and b-file from N. J. A. Sloane, Apr 08 2016 using data from A003022.

A325768 Number of integer partitions of n for which every restriction to a subinterval has a different sum.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 12, 15, 15, 23, 22, 29, 32, 40, 42, 55, 56, 71, 75, 92, 100, 124, 128, 152, 167, 198, 212, 255, 269, 315, 343, 392, 428, 501, 529, 615, 665, 757, 812, 937, 1002, 1142, 1238, 1385, 1490, 1701, 1808, 2038, 2200, 2476

Offset: 0

Gus Wiseman, May 21 2019

nonn

Also the number of Golomb rulers of length n whose consecutive marks are separated by weakly decreasing distances.

The Heinz numbers of these partitions are given by A325779.

			The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)
                        (41)  (51)  (52)   (62)   (63)
                                    (61)   (71)   (72)
                                    (421)  (521)  (81)
                                                  (432)
                                                  (531)
                                                  (621)

Cf. A000041, A002033, A103300, A143823, A169942, A325676, A325687, A325769, A325779.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,0,30}]

A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 0, 1, 1, 4, 4, 0, 1, 1, 5, 5, 0, 0, 1, 1, 6, 12, 4, 0, 0, 1, 1, 7, 12, 5, 0, 0, 0, 1, 1, 8, 25, 8, 4, 0, 0, 0, 1, 1, 9, 24, 12, 3, 0, 0, 0, 0, 1, 1, 10, 40, 32, 8, 4, 0, 0, 0, 0, 1, 1, 11, 41, 41, 6, 3, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, May 13 2019

A composition of n is a finite sequence of positive integers summing to n.

			The distinct consecutive subsequences of (1,1,3,3) are (1), (1,1), (3), (1,3), (1,1,3), (3,3), (1,3,3), (1,1,3,3), all of which have different sums, so (1,1,3,3) is counted under a(8).
Triangle begins:
  1
  1  1
  1  2  1
  1  3  0  1
  1  4  4  0  1
  1  5  5  0  0  1
  1  6 12  4  0  0  1
  1  7 12  5  0  0  0  1
  1  8 25  8  4  0  0  0  1
  1  9 24 12  3  0  0  0  0  1
  1 10 40 32  8  4  0  0  0  0  1
  1 11 41 41  6  3  0  0  0  0  0  1
  1 12 60 76 14  4  4  0  0  0  0  0  1
  1 13 60 88 16  6  3  0  0  0  0  0  0  1
Row n = 8 counts the following compositions:
  (8)  (17)  (116)  (1115)  (11111111)
       (26)  (125)  (1133)
       (35)  (143)  (2222)
       (44)  (152)  (3311)
       (53)  (215)  (5111)
       (62)  (233)
       (71)  (251)
             (332)
             (341)
             (512)
             (521)
             (611)

Row sums are A325676.
Column k = 2 is A000027.
Column k = 3 is A325688.
Cf. A000079, A007318, A048004, A108917, A143823, A169942, A266223, A325592, A325680, A325685.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,15},{k,n}]

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 5, 4, 5, 5, 5, 2, 4, 4, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 2, 4, 4, 6, 3, 7, 7, 7, 4, 7, 4, 7, 7, 7, 7, 7, 4, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 8, 6, 8, 8, 8, 4, 8, 6, 8, 6, 8, 8

Offset: 0

Gus Wiseman, Jun 02 2020

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 5 4 5 5 5
  2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
  2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7

Row lengths are A011782.
Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Golomb rulers are counted by A169942 and ranked by A333222.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702
Knapsack compositions are counted by A325676 and ranked by A333223.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Knapsack compositions are counted by A334268 and ranked by A334967.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Total/@Subsets[stc[n]]]],{n,0,100}]

a(n) = A299701(A333219(n)).

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 5, 5, 4, 1, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 5, 1, 3, 3, 5, 2, 6, 6, 7, 3, 6, 3, 8, 6, 7, 8, 9, 3, 5, 6, 8, 6, 8, 7, 11, 5, 8, 8, 11, 7, 11, 9, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 9, 5, 9, 9, 9, 3, 6, 5, 9, 5, 7, 8, 11, 6, 9, 8, 11, 9, 11, 11, 11, 3, 5, 6, 8, 5, 9

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

A325769 Number of integer partitions of n whose distinct consecutive subsequences have different sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 62, 61, 88, 87, 123, 121, 168, 164, 234, 225, 306, 306, 411, 401, 527, 533, 700, 689, 894, 885, 1163, 1150, 1452, 1469, 1866, 1835, 2333, 2346, 2913, 2913, 3638, 3619, 4511, 4537, 5497, 5576, 6859, 6827, 8263

Offset: 0

Gus Wiseman, May 21 2019

nonn

For example (3,3,1,1) is counted under a(8) because it has distinct consecutive subsequences (), (1), (1,1), (3), (3,1), (3,1,1), (3,3), (3,3,1), (3,3,1,1), all of which have different sums.

The Heinz numbers of these partitions are given by A325778.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

Cf. A000041, A002033, A143823, A169942, A325676, A325677, A325683, A325768, A325770, A325778.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,30}]

a(41)-a(53) from Fausto A. C. Cariboni, Feb 24 2021

A325680 Number of compositions of n such that every distinct circular subsequence has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 14, 16, 29, 24, 42, 46, 78, 66, 146, 133, 242, 208, 386, 352, 620, 494, 948, 842, 1447

Offset: 0

Gus Wiseman, May 13 2019

A composition of n is a finite sequence of positive integers summing to n.

A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (222)     (61)       (62)
                                     (111111)  (124)      (71)
                                               (142)      (125)
                                               (214)      (152)
                                               (241)      (215)
                                               (412)      (251)
                                               (421)      (512)
                                               (1111111)  (521)
                                                          (2222)
                                                          (11111111)

Cf. A000079, A008965, A108917, A143823, A169942, A276024.

Cf. A325545, A325676, A325682, A325685, A325687, A325688.

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@subalt[#]&]],{n,0,15}]

a(18)-a(25) from Robert Price, Jun 19 2021

A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 2, 2, 8, 17, 26, 43, 77, 129, 210, 351, 569

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
  ()  (1)  .  .  (22)  (122)  (1122)  (11221)  (21122)  (333)
                       (221)  (1221)  (12211)  (22112)  (22113)
                              (2211)                    (22122)
                                                        (31122)
                                                        (121122)
                                                        (122112)
                                                        (211221)
                                                        (221121)
For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).

The version for partitions is A325702.
The recursive version is A353391, ranked by A353431.
The consecutive case is A353392, ranked by A353432.
These compositions are ranked by A353402.
The reverse version is A353403.
The recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
A353400 counts compositions with all run-lengths > 2.
Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#],Length/@Split[#]]&]],{n,0,15}]

A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 5, 3, 9, 1, 9, 5, 7, 5, 11, 1, 13, 5, 9, 5, 13, 3, 13, 7, 9, 5, 17, 1, 17, 5, 9, 9, 15, 5, 15, 5, 13, 5, 21, 1, 17, 9, 9, 9, 17, 3, 21, 7, 13, 5, 17, 5, 21, 9, 13, 5, 21, 1, 21, 9, 11, 13, 19, 5, 17, 5, 17, 5, 29, 1, 21, 9, 9, 13, 17, 5, 25, 7, 17, 7

Offset: 0

Gus Wiseman, May 13 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Compare to the definition of perfect partitions (A002033).

			The distinct consecutive subsequences of (3,4,1,1) together with their sums are:
   1: {1}
   2: {1,1}
   3: {3}
   4: {4}
   5: {4,1}
   6: {4,1,1}
   7: {3,4}
   8: {3,4,1}
   9: {3,4,1,1}
Because the sums are all different and cover {1...9}, it follows that (3,4,1,1) is counted under a(9).
The a(1) = 1 through a(9) = 9 compositions:
  1   11   12    1111   113     132      1114      1133       1143
           21           122     231      1222      3311       1332
           111          221     111111   2221      11111111   2331
                        311              4111                 3411
                        11111            1111111              11115
                                                              12222
                                                              22221
                                                              51111
                                                              111111111

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Fausto A. C. Cariboni, All compositions that yield a(n) for n = 1..100, Feb 21 2022.

Cf. A000079, A002033, A103295, A126796, A143823, A169942, A325676, A325677, A325683, A325684.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Sort[Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]]==Range[n]&]],{n,0,15}]

a(21)-a(25) from Jinyuan Wang, Jun 26 2020

a(21)-a(25) corrected, a(26)-a(80) from Fausto A. C. Cariboni, Feb 21 2022

cp's OEIS Frontend

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

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A143824 Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.

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A325768 Number of integer partitions of n for which every restriction to a subinterval has a different sum.

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A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.

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A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

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A124770 Number of distinct nonempty subsequences for compositions in standard order.

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A325769 Number of integer partitions of n whose distinct consecutive subsequences have different sums.

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A325680 Number of compositions of n such that every distinct circular subsequence has a different sum.

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